Topological Quantization of Complex Velocity in Stochastic Spacetimes

Abstract

We establish a rigorous geometric framework for quantum fields on a stochastic gravitational background. Starting from a master partition function that averages over metric fluctuations, we define a matter amplitude $\mathcal{K}$, whose logarithmic derivative yields a complex velocity field $\hat{I}{\cdot }_{\hat{I}}¼=\ddot{I}{€}_{\hat{I}}¼-i{u}_{\hat{I}}¼$. This object, originating in Nelson's stochastic mechanics, is a section of the pullback bundle $E=\ddot{I}{€}_2^{\ast }(T^{\ast }M)$ over the product of configuration space $\mathcal{C}$ and spacetime $M$. We prove that $\hat{I}{\cdot }_{\hat{I}}¼$ defines a flat $U(1)$ connection with $\mathcal{K}$ as its horizontal section, and via a bundle isomorphism it maps to the symmetric logarithmic derivative of quantum estimation theory. The coupled dynamics collapse into ${\mathcal{L}}_{\hat{I}}\cdot \hat{I}\cdot =d(|\hat{I}\cdot {|}^2)$. We resolve the tension between flatness and multi-valuedness: although the connection is flat, the potential can be multi-valued from topological terms or branch cuts. The total phase satisfies $\frac{m}{\hslash }{\oint }_{\hat{I}}^3\hat{I}{\cdot }_{\hat{I}}¼d{x}^{\hat{I}}¼=2\ddot{I}€n+\hat{I}”\ddot{I}{•}_{\text{top}}$. We demonstrate this in a toy model: a scalar field on a conical spacetime with deficit angle $\hat{I}\pm$, computing the matter amplitude in the Gaussian approximation, deriving the complex velocity, and calculating its holonomy. The resulting topological offset receives a quantized stochastic correction depending on the variance of metric fluctuations, providing an experimental signature for atom interferometry. This framework geometrizes quantum mechanics without hidden variables: stochasticity imprints spacetime fluctuations on matter, preserving the wave function's probabilistic nature while giving a geometric origin for the Born rule.